Advanced calculus

Elementary set theory

A set is a collection of objects, usually defined by a common defining property.

Set operations

• Subset: `B sub A` if `x in B => x in A`
• Union: `A uu B = { omega : omega in A or omega in B }`
• Intersection: `A nn B = {omega: omega in A and omega in B}`
• Complement: `A^c = {omega: omega !in A}`
• Symmetric difference: `A delta B = (A^c nn B) uu (A nn B^c)`
• Cartesian product: `A xx B` is all ordered pairs `(omega_1 in A, omega_2 in B)`
• Product space: Let `{Omega_alpha : alpha in I}` be a collection of nonempty sets, then `X_(alpha in I} Omega_alpha` is the product space defined as `{f : f " defined on" I "st" AA alpha f(alpha) in Omega_alpha}`. Axiom of choice states that this space is non-empty

Equivalence relationships

Let `G` define an relationship (`G sub Omega xx Omega`). It is an equivalence relationship if it is :

• reflexive: `AA x in Omega, (x,x) in G`
• symmetric: `x ~ y => y ~ x`
• transitive: ` x ~ y, y~z => x~z`

`[x]` defines the equivalence class `{y: x~y}`. Equivalence classes form a partition over `Omega`

Functions

A function is a correspondence between elements in a set `X` (domain) and `Y` (range). `AA x in X` there is a unique `y in Y` that corresponds to it: `y = f(x)`.

• onto: if `f(X) = Y`
• one to one: if `AA y in f(X) EE "a unique" x in X`
• if `f` is onto and one-to-one then `X` and `Y` have same cardinality
• monotone: if `x > y => f(x) > f(y)`

A set `X` is:

• finite if `EE n in NN` st `X` and `Y = {1,2, ..., n}` have the same cardinality
• countable if `X` and `NN` have the same cardinality
• uncountable if not finite or countable

Induction

The set `NN` of natural numbers has the well ordering property such that nonempty subset of `NN` has a smallest element `s` st `s in A` and `a in A => a >= s`. Principle of induction is consequence of this property, and it states:

Let `{ P(n) : n in NN}` be a collection of propositions which are either true or false. If `P(1)` is true and `P(n) "is true" => P(n+1)` then `P(n)` is true for all `n`.

Real numbers

At least two approaches to defining the real numbers:

Start with `NN`, then construct `Z = ( NN uu {0} uu (- NN) )`, then `QQ`, then `RR` as either the set of all Cauchy sequences of rationals or as Dedekind cuts.

OR, can define the real numbers as the set that satisfies three axioms: `RR` is a field (algebraic), ordered and complete, ie (complete ordered field).

Algebraic axioms:

• commutative group under +
• multiplicative group under `**` (excluding 0)
• distributive axioms

Order axiom states that there is set `P sub R` (positive numbers) st:

• `x, y in P => xy in P, x+y in P`
• `x in P => -x !in P`
• `x in R => x = 0 or x in P or x in -P`

Given such a `P`, can define an order on `R` by defining `x < y` to mean `x-y in P`, and it follows that `AA x, y in R`, either `x=y, or x lt y or x > y` (linear order)

Let `A sub R`. `M` is an upperbound of `A` if `a in A => a <= M`. supremum `A` means the least upperbound of `A` (`sprA` is an upperbound for `A`, but if `K < sprA` then `K` is not an upperbound for `A`)

• `a in A => a <= sprA`
• `K < sprA => EE a in A "st" K < a`

Completeness axiom states that if `A sub R` has upperbound `< oo` then `EE stackrel~M < oo "st" stackrel~M = sprA`. Consequences: (Axiom of Eudoxus) `AA x in R EE n in NN "st" x < n`; `Q` is dense in `R`.

Extended real numbers `barR = {-oo, R, +oo}` and extend definitions of `+, *, <` intuitively.

Sequences and limits

A sequence of real numbers is the range of a function `f: NN -> RR`

`lim_n x_n = a in R if AA epsilon > 0 EE n_epsilon "st" n >= N_epsilon => |x_n - a| < epsilon`

Let `{x_n} sub R`, `y_n = spr{x_j : j >= n}` then `y_n` is a non-decreasing sequence, and can be shown that all monotone nonincreasing sequences in R converge to `+oo` or a finite real number, so `lim_n y_n = y` exists and is called lim sup and written `bar{lim_n} x_n = inf_n (spr_(j>=n) x_j)`.

An alternative definition of `bar(lim_n)` is `bar(lim_n) = a in R if AA epsilon > 0 EE N_epsilon "st" n > N_epsilon => x_n < a + epsilon, and AA k EE n >= > k "st" x_n > a-epsi`.

A sequence `{x_n} in R` is a Cauchy sequence if `AA epsilon > 0, EE N_epsilon "st" n,m >= N_epsilon => |x_m - x_n| < epsilon`

Prop: if `{x_n} in R` is convergent in `R <=> {x_n}` is Cauchy. (proof requires completeness axiom).

Series

Let `{x_n}_{n>=1}` be a sequence of real numbers. Let `s_n = sum_{j=1}^n x_j, n=> 1` be the nth partial sum. The series `sum_1^oo x_j` converges to `s` in `R` if `lim_n s_n = s`. If `x_j >= 0` then `lim_n s_n = s in R or oo`. A series `sum_1^oo x_j` converges absolutely if `sum_1^oo |x_j|` converges. A series converges uniformly if its seuqence of partial sums converges uniformly.

Power series

Let `{a_n}_(n=>0)` be sequence of real numbers. The series `sum_1^oo a_n x^n` is called a power series and is said to be convergent on `B` if it converges `AA x in B`.

The radius of convergence `R = (barlim |a_n|^(1/n))^-1`. Power series converges when `|x| < R` and diverges to `oo` when `|x| > R`.

Taylor Series

`f: (a - eta, a + eta) -> RR, a in RR`. Suppose `f` is `n` times differentiable in `(a - eta, a+eta)`. Let `a_n = (f^(n)(a))/(n!)`. Then the power series `sum a_n (x- a)_n` is called the Taylor series of `f` at `a`. Let `a_n = (f^(n)(a))/(n!)`, then the power series `sum a_n(x-a)^n` is called the Taylor series of f. The Taylor remainder theorem says `AA x in I` and any `n >= 1' `|f(x) – sum a_j x^{j| <= |f}((n+1))(y_n)|/((n+1)!) |x-a|^{` for some `y_n in I`. So if `lambda_k = sum |f}(k)(y)|` and `sum lambda_k/(k!)` converges then the remainder `-> 0`.

Metric spaces

Let `S != O/` and `d: S x S -> R^+` then if:

• `d(x,y) = d(y,x)`
• `d(x,z) <= d(x,y) + d(y,z)`
• `d(x,y) = 0 iff x=y`

then `d` is called metric on `S` and the pair `(d,S)` is a metric space

A sequence `{x_n} sub (S, d)` converges to `x in S` if `AA epsilon > 0 EE N_epsilon "st" n => N_epsi => d(x_n, x) < epsi` and is written `lim_n x_n = x`. A sequence in a metric space is Cauchy if `AA epsi > 0 EE N_epsi "st" n,m >= N_epsi => d(x_n, x_m) < epsi`. (any convergent sequence is Cauchy). A metric space is complete is every Cauchy sequence converges.

Continuous functions

Let `(S,d) and (T, rho)` be two metric spaces and `f:S -> T` a map between them. `f` is:

• continuous at a point, `p in S` if `AA epsi > 0 EE delta > 0 "st" d(x,p) < delta_epsi => rho(f(x), f(y)) < epsi`,
• continuous on a set, `B` if continuous at every `b in B`,
• uniformly continuous if `AA epsi > 0 EE delta > 0 "st" AA x,y in S, d(x,y) < delta => rho(f(x), f(y)) < epsi`

`O sub (S,d)` is open if `x in O => EE delta > 0 "st" d(x,y) < delta => y in O`. ie. at every `x in O` an open ball `B_x(delta)` of positive radius `delta` is `sup O`. A set is closed if its complement is open.

A map `f: S -> T` is continuous iff `AA O` open in `T` `f^{-1}(O)` is open in `(S,d)`

A collection of open sets `{O_alpha : alpha in I}` is an open cover for a set `B sub (S,d)` if `AA x in B, EE a in I "st" x in O_alpha`.

A set is compact if a finite subcollection of an open cover is an open cover.

Any `K sub R` is compact iif it is bounded and closed.